Zum Zeitverhalten des Einfangs diffundierender Teilchen auf Gittern mit statistisch verteilten Haftstellen

Anlauf, J. K.

Book/Report

FZJ-2018-03092

Zum Zeitverhalten des Einfangs diffundierender Teilchen auf Gittern mit statistisch verteilten Haftstellen

Anlauf, J. K. (Corresponding author)FZJ*

1989
Kernforschungsanlage Jülich, Verlag Jülich

Jülich : Kernforschungsanlage Jülich, Verlag, Berichte der Kernforschungsanlage Jülich 2259, 131 p. (1989)

Please use a persistent id in citations: http://hdl.handle.net/2128/18613

Report No.: Juel-2259

Abstract: The trapping of diffusing particles on lattices with a random distribution of static traps is studied. A single particle performs a random walk in d dimensions and getstrapped immediately when it encounters a trap site for the first time. The traps are randomly distributed with concentration c. Some of the quantities studied in this thesis are the survival probability, the mean number of distinct sites visited and the mean square displacement after n steps. Asymptotic results for small and large step numbers are well known. In this thesis the crossover between the two regimes is studied in full detail for random walks on three dimensional lattices with a new Monte Carlo simulation method. The contents of the thesis are as follows : After an introduction in part I a detailed definition of the trapping problem is given in part II. For this purpose a discussion of the averaging process is needed, because two different types of randomness (random walks and statistically distributed traps) are involved. The known asymptotic results for the trapping problem are also collected in this part. Finally an exact correspondence between the trapping problem and the self-interacting-walk model (SIW) is described. In part III an exact solution in one dimension is found for a correlated walk, where the probability of a step into a given direction depends on the direction of the last step performed. Asymptotically exact results are given for the distribution function for the number of distinct sites visited on a linear chain without any traps and for the survival probability on a chain with statistically distributed traps. These formulas include numerous correction terms to the leading behavior. Also an exact formula is given for the mean number of steps needed to find a site that was not visited before. In leading order some interesting scaling laws are found for this type of walk. In higher dimensions no exact solution seems possible, but Monte Carlo simulations can be done to study the trapping problem. In part IV earlier simulation methods are analyzed. Since all are shown to have severe problems to reach the asymptotic regime of large step numbers, a new method is developed in this part of the thesis. It uses importance sampling to simulate effectively those walks that contribute most to the survival probability, and can realize much better statistics than any other known simulation method. To get small correlation times in the simulation a special algorithm is used that does not use a detailed balance condition. With the help of hash-coding technics and a heap (although implemented in FORTRAN) many walks can be simulated simultaneously. This makes large parts of the algorithm vectorizable and guarantees an effective use of vector computers (such as the CRAY X-MP in the Kernforschungsanlage Jülich). The simulation results in three dimensions are given in part V, and scaling theories are used to analyze them. The crossover takes place at much lower step numbers and higher survival probabilities than expected previously, and can best be studied if one looks at the mean number of distinct sites visited in the presence of traps. There the crossover is much sharper than for the survival probability. A new and simple approximation for the survival probability in two and more dimensions emerges from the observed crossover behavior that is valid for all step numbers. The mean square displacement of surviving particles in the presence of traps shows the surprising result that, for small trap concentrations, there is a range of step numbers where the mean square displacement becomes smaller when the step number is increased. The final part VI contains a summary in German and some concluding remarks. Although the survival probability at the crossover point is much higher than expected, it is still too small to be measured in present experiments. Moreover, even macroscopic lattices with 6 $\cdot$ 10$^{23}$ atoms are to small to show the necessary fluctuations in trap concentration that are needed to see the asymptotic stretched exponential decay of the survival probability, if the trap concentration is below 3%.

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